Abstract
In this paper we explore the construction of arbitrarily tight alpha BB relaxations of C^2 general non-linear non-convex functions. We illustrate the theoretical challenges of building such relaxations by deriving conditions under which it is possible for an alpha BB underestimator to provide exact bounds. We subsequently propose a methodology to build alpha BB underestimators which may be arbitrarily tight (i.e., the maximum separation distance between the original function and its underestimator is arbitrarily close to 0) in some domains that do not include the global solution (defined in the text as “sub-optimal”), assuming exact eigenvalue calculations are possible. This is achieved using a transformation of the original function into a mu -subenergy function and the derivation of alpha BB underestimators for the new function. We prove that this transformation results in a number of desirable bounding properties in certain domains. These theoretical results are validated in computational test cases where approximations of the tightest possible mu -subenergy underestimators, derived using sampling, are compared to similarly derived approximations of the tightest possible classical alpha BB underestimators. Our tests show that mu -subenergy underestimators produce much tighter bounds, and succeed in fathoming nodes which are impossible to fathom using classical alpha BB.
Highlights
In this paper we discuss the problem of locating a global minimum of a general C2 non-linear function f : X → F ⊂ R, X = {x : x ∈ [x L, xU ]} ⊂ RN, denoted as problem P: This article is dedicated to the memory of Christodoulos A
Modern deterministic global optimization methods guarantee that the global optimum of many general C2 optimization problems can always be located with certainty, and in finite time, this time may be too long to be viable in practice
The purpose of this work is to prove that it is possible to achieve arbitrarily tight αBB underestimators, but we would like to stress that this is a theoretical proof-of-concept: in practice, there is no way to obtain exact eigenvalue bounds for a general function, and we investigate the computational behaviour of the approach based on eigenvalue sampling, similar to [17,57], in some domains that do not contain the global solution
Summary
The purpose of this work is to prove that it is possible to achieve arbitrarily tight αBB underestimators, but we would like to stress that this is a theoretical proof-of-concept: in practice, there is no way to obtain exact eigenvalue bounds for a general function, and we investigate the computational behaviour of the approach based on eigenvalue sampling, similar to [17,57], in some domains that do not contain the global solution The results of this investigation are in line with our theoretical predictions, as using this heuristic approach we are able to fathom nodes which may not be fathomed using classical αBB.
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